Biaxially twisted structure

ABSTRACT

The present invention provides a biaxially twisted structure having curved developable surfaces which can be represented by a simple net. The net is in a two-dimensional shape of two congruent rotationally symmetric convex polygons misalignedly connected along a shared edge. A three-dimensional shape of the structure can be reconstructed from the net by sequentially connecting its edges. The structure has a unique design and mechanical feature, which can be utilized as a box, container, part, toy, building and so on.

TECHNICAL FIELD OF THE INVENTION

The invention relates to a twisted structure having curved surfaces that can be represented by a simple net.

BACKGROUND OF THE INVENTION

A basic geometric shape of a typical box is a prism. In particular, a cuboid is practical since it can be represented by a simple net, the structure is stable, and it fills a space in three orthogonal directions. Other basic geometric shapes such as an antiprism are rarely used, and thus diversities are limited. Also, the shapes of boxes are mostly polyhedral, consisting of flat faces, while those having curved faces are few. Moreover, there is a limited way to create curved surfaces in boxes, primarily by changing a straight edge into a curved edge to create curved surfaces.

SUMMARY OF THE INVENTION

The present invention, therefore, provides a novel structure with an attractive design and functional features that can be used as a box, container, toy, part, building and so on. A shape of the twisted structure is a curved body, twisted in two orthogonal axes, and can be represented by a simple net, which is different from common basic shapes like a prism or antiprism.

In one aspect, the present invention provides a structure, having a three-dimensional shape that can be represented by a net in a two-dimensional shape, the two-dimensional shape of the net including: 2 congruent rotationally symmetric convex polygons, wherein the polygon is a n-gon and n is an integer more than 3; 2n three-valent main-vertices, wherein each of the main-vertices is placed at each corner of the polygons without overlapping; 2n three-valent sub-vertices, wherein n of the sub-vertices are placed at rotationally symmetric positions within each of the polygons without overlapping; 2n main-edges, wherein each of the main-edges is straight, and each of the main-edges connects 2 of the main-vertices without crossing; and 4n sub-edges, wherein each of the sub-edges is either straight or curved, each of the sub-edges connects two of the sub-vertices or connects one of the main-vertices and one of the sub-vertices, and 2n of the sub-edges are placed at rotationally symmetric positions within each of the polygons without crossing, wherein: each of the main-vertices contacts 2 of the main-edges and one of the sub-edges, and each of the sub-vertices contacts 1 main-edge and the 2 sub-edges, and wherein the structure has curved developable surfaces with a biaxially twisted appearance. The net is simple but structure represented by the net is rather complex. The main-edges form a basic skeleton of the structure, and the sub-edges expand the degree of freedom in creating new designs. Incorporation of regular polygons in the net endows the structure with graceful appearance.

In another aspect, the present invention provides said structure, wherein the structure is a box and the net further includes a plurality of flaps and/or crease lines. A box of this new design can be made without using tools or other elements.

In a further aspect, the present invention provides said structure, wherein the structure rotates in a flow about its rotational axis when a direction of the rotational axis is in parallel to a direction of the flow. Providing a shaft, the structure rotates in a stream like a pinwheel.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a perspective view of a square twisted structure;

FIG. 2 shows a square twisted structure and a Penrose square;

FIG. 3 shows nets and subtypes of the square twisted structures;

FIG. 4 shows plan and sectional views of regular-polygonal twisted structures;

FIG. 5 shows a rectangular twisted structure;

FIG. 6 is an example of designing the square- and rectangular twisted structures;

FIG. 7 is an example of designing the square twisted structures with linear or curved sub-edges;

FIG. 8 is an example of the square twisted box with flaps and so on;

FIG. 9 is an example of the square twisted structure that rotates in a flow; and

FIG. 10 is an example of the rectangular twisted structure making awkward movements.

DESCRIPTION OF THE EMBODIMENTS

The invention was conceived during analyses of a number of polyhedra with various basic geometric shapes. In particular, the idea was developed from a certain kind of polyhedra, consisting of only 4-gons (i.e. tetragons) with 3- and 4-valent vertices (e.g. “3-valent vertex” is a vertex with 3 edges), among those homeomorphic to a sphere. The structure of this invention, however, is not a polyhedron, but a curved body made of developable surfaces (“developable surface” is a surface that can be flattened onto a plane without stretching or compressing).

The structure described herein is complex but it can be represented by a simple net (“net” is a two-dimensional shape obtained by flattening a three-dimensional shape). It has a basic geometric shape different from a prism or antiprism. The structure is twisted in two orthogonal axes and may provide an elegant exterior. The top and a bottom faces are near-flat, which may be stably placed on a level surface. A square, rectangular, or hexagonal type resembles a space-filling solid, and is suitable for storing. Furthermore, the structure has unique features, in which it rotates in a stream or displays awkward motions under defined conditions.

FIG. 1 shows a perspective view of a square twisted structure. Edges of the structure are indicated by solid lines (visible parts) and dashed lines (hidden parts). The structure is a curved body but not a polyhedron. An exterior of the structures is regularly skewed. The twists are seen in both horizontal and vertical directions. For the twist in the horizontal direction (left-right), the center areas of the top and bottom faces are twisted in the opposite direction about a rotation axis vertically passing through the centers of the structure. For the twit in the vertical direction (top-bottom), the surrounding area of the top and bottom faces are locally twisted in the opposite direction about a rotation axis horizontally extending from the center of the structure to a corner or side of the structure (in the square twisted structure, opposing corners or sides in the surrounding area are twisted in the opposite direction). The center areas of the top and bottom faces are about flat, allowing the structure to be placed stably on the level surface. In contrast, the surrounding area the top and bottom faces is undulating dynamically. Looking from the side, an interfacial ridge between the top and bottom faces curvilinearly links one vertex on the bottom face to the other vertex on the top face.

This structure may look similar to an antiprism, but these are intrinsically different. The antiprism is a polyhedron: 1) all edges are straight, 2) all faces are planar, 3) all vertices are 4-valent, 4) the twist is one-axial (horizontal axis), and 5) side faces are triangles. The twisted structure, on the other hand, is a curved body: 1) edges are curved, 2) the faces are curved, 3) the vertices are basically 3-valent, 4) the twist is bi-axial (horizontal and vertical axes), and 5) side faces are pentagons (tetragon-like pentagons, in some cases). However, as described later, an antiprism may be obtained as an ultimate shape upon transformation of the twisted structure.

FIG. 2 left is a plan view of the square twisted structure with its edges (solid lines for visible parts, dashed lines for hidden parts). The center areas of the top and bottom faces (in this example, the center area is a square) are tilted. The center area is almost planar. The surrounding area is inclined toward the outside, and the angle varies depending on the location. Interestingly, many of the straight edges in the net (described later) transform into curved edges in the twisted structure. This is essentially different from a conventional method to create curved surfaces by introducing curved edges into the net. In FIG. 2 right, an impossible figure “Penrose square”, is shown. As compared to the twisted structure, the surrounding areas look alike, but connections of the inner edges are different.

FIG. 3 shows an exemplary net of the square twisted structure. The net, set forth in elementary geometry, is a two-dimensional representation of a three-dimensional shape, and does not include additional elements such as flaps. For example, the net of a cube can be drawn as a planar arrangement of six congruent squares. However, the net in this context encompasses its equivalents that represent the same three-dimensional shape. For example, the cube can be represented by 11 different nets and all of which are deemed equivalent in this context.

The net of the twisted structure has a shape of two congruent rotationally symmetric convex polygons which are connected along a shared edge with a parallel displacement along the shared edge (call it “misalignedly connected”). Here, “congruent” means a state of polygons having the same shape and size that perfectly overlap each other. “rotational symmetry” refers to symmetry that the original figure perfectly overlaps with that rotated by an angle of 360/n degrees (n-fold symmetry, wherein n is an integer greater than or equal to 2) about the rotation axis. A “convex polygon” is a polygon with every interior angle of less than 180 degrees and each edge being a straight line segment. Taking the net of the square twisted structure as an example, the shape of the polygon is a square, which is convex (every internal angle is 90 degrees) and rotationally symmetric (4-fold symmetry), and two congruent squares are misalignedly connected along the shared edge. This looks simpler than a typical net of a prism.

There are several ways to misalignedly connect two congruent convex polygons. FIG. 3 illustrates three exemplary configurations of two squares: translated upwardly (center top), downwardly (top left), and to the midpoint (top right). Shown below are the plan views of their exemplary three-dimensional structures. The left and middle nets are mirror-image as two convex polygons are displaced by the same width, and resulting three-dimensional structures are chiral as well (as mirror-image isomer). For paper models, mirror-image isomers are readily obtained from the same net by either mountain-folding or valley-folding. This is not the case for the net on the right, in which its mirror-image net leads to the same three-dimensional structure and thus has no mirror-image isomer. In this case, the center area of the top and bottom faces is no longer available.

As shown in FIG. 4 , the embodiment encompasses twisted structures based on other regular polygons such as regular pentagon, regular hexagon, and so on. Looking at the cross section, the side of the twisted structure is tapered or “V-shaped”. This bent forms an interface at which the top and bottom faces meet. An angle at the bent is about constant along the edge. The approximate angle δ is 90° for square, 76.3° degrees for pentagonal, and 70.5° degrees for hexagonal twisted structures (180° for triangular), based on a formula of dihedral angle.

The twisted structure can be designed based on other convex polygons. FIG. 5 shows a net of a rectangular twisted structure. The center area of the obtained structure is approximately flat, but curvature is more apparent than that of the square twisted structure.

FIG. 6 shows a more detailed net of the square and rectangular twisted structures. For congruent convex n-gons (n is an integer greater than or equal to 4), the net has 2n primary vertices (main-vertices) at corners of the polygons, which constitute important vertices in the skeleton of the twisted structure. The main-vertices in the net should be positioned properly so as not to overlap each other upon reconstruction of its three-dimensional structure. The tetragonal twisted structure has 8 main-vertices (from 1 to 8), in which a tetragon 1357 forms the top face, a tetragon 2468 forms the bottom face. There are also 2n primary edges (main-edges), each making a connection between two main-vertices. The tetragonal twisted structure has 8 main-edges (solid lines in the net, 12, 23, 34, 45, 56, 67, 78, and 81).

In addition to the main-vertices, 2n secondary vertices (sub-vertices) may be chosen at rotationally symmetric positions within the convex polygons in the net. Sub-vertices expand the degree of freedom in designing. In FIG. 6 , the tetragonal twisted structure has 8 sub-vertices (from a to h: a-d within the top face, e-h within the bottom face). Each sub-vertex is 3-valent, connecting to one main-vertex (e.g. a to 2) and to two other sub-vertices (e.g. a to b and a to d). In this operation, the tetragonal twisted structure gains 4n secondary edges (sub-edges) which can be either straight or curved, provided that the positions of sub-edges are rotationally symmetric relative to the corresponding convex polygons in the net. In the examples of FIG. 6 , there are 16 sub-edges (dashed lines in the net, 8 sub-edges in the top face [ab, bc, cd, da, a2, b4, c6, and d8], 8 sub-edges in the bottom face [ef, fg, gh, he, e1, f3, g5, and h7]) and all are straight edges. The center area and surrounding areas described above can be redefined using the sub-vertices. For example, in the top face of the twisted structure in FIG. 6 , the center area is the area enclosed by the sub-vertices abcd, and the surrounding area is the area outside the center area.

Taking the square twisted structure as an example, the approximate height H and twist angle (each for the top or bottom center area) ε are calculated. To simplify, sub-edges are made in parallel to main-edges. A is the offset width of two squares and B is the width of the center area.

$\begin{matrix} {H = {A\sqrt{\left\lbrack {2 - {2{A^{2}/\left( {{2A} + B} \right)^{2}}}} \right\rbrack}}} & \left\lbrack {{Math}1} \right\rbrack \end{matrix}$ $\begin{matrix} \left. {\varepsilon = \left( {{18\text{?}{arc}\text{?}2A} + B} \right)} \right\rbrack & \left\lbrack {{Math}2} \right\rbrack \end{matrix}$ ?indicates text missing or illegible when filed

Rectangular twisted structures is complex. The center area is approximately flat, but curvature in the diagonal direction is more evident depending on aspect ratios. Heights at the long and short sides are different, where the long side is higher, making it less stable as compared to the square-type. However, it creates unique motion including swaying back and forth when placed on a level surface.

FIG. 7 illustrates rearrangements of sub-vertices and sub-edges in the square twisted structure, while preserving rotational symmetry. In the top figure, sub-vertices are repositioned but sub-edges remain straight. The center area is a nearly flat square and the surrounding areas consist of curved pentagonal surfaces. This object is like a propeller, and functionally attractive. In the bottom figure, sub-vertices are repositioned and also sub-edges are made into arcs. The center area is a hyperbolic square with curvatures around corners. It is a star-like and visually pleasing. Such transformation may be useful to design objects for structural or functional appreciation.

A transformation of the twisted structure ultimately leads to an antiprism. The transformation is achieved by shifting the position of each sub-vertex to the main-vertex connected (e.g. a->2, b->4, c->6, and d->8 for the top face, e->1, f->3, g->5, and h->7 for the bottom face in FIG. 6 ). As a result, sub-faces that make up the surrounding area or side of the twisted structure change from pentagons to triangles, and the main-vertices change from 3-valent to 4-valent.

All the twisted structure as shown in FIG. 1-10 have been produced using a paper or plastic sheet.

The twisted structure has curved surfaces. When materials such as a paper and plastic sheet are used (substantially flat and flexible, but relatively hard to stretch), structural strains will be generated as curved surfaces are formed. Such strains might destabilize curved structures as compared to polyhedral structure. However, necessary measures known in the art (such as adhesion and hooking) provide sufficient strength.

For other materials, the twisted structure are produced only to have a “twisted appearance” without introducing unnecessary strains. In this case, three-dimensional data are useful, which are obtained by computer modeling (e.g. CAD modeling or theoretical calculation based on the net) or actual modeling (e.g. 3D scanning of a prototype obtained from the net). For production, various methods known in the art may be employed (molding, 3D printing, NC processing, etc.). For example, containers may be produced by molding (e.g. injection molding of plastic resins), and parts may be produced by 3D printing or NC processing. For buildings, skeletal structures described herein are used as primary frames, which may be strengthened with secondary frames by triangulating the primary frames.

FIG. 8 shows an example of nets to make a box, by incorporating additional elements such as flaps. The box can be produced by directly folding materials (such as paper, plastic sheets, and the like) in the shape of such nets. A mirror-image structure can be made by changing mountain-folding to valley-folding, and vice versa. Crease lines along the edges ease folding. Flaps enables ones to form a stable structure without using tools or other materials (top left). Here, the structure is stabilized by corner flaps 10 that support box corners (near the main-vertices) and side flaps 11 that support box sides. The structure can be further stabilized by adhesion (bottom left, flap 12 for adhesion) or hooking (top right, insert a tab T into a slit S).

A regular-polygonal twisted structure resembles a pinwheel or propeller. In fact, providing a shaft to the axis of rotation, the structure turns as you blow on. FIG. 9 shows the movement of the square twisted structure in a flow. The structure is positioned in the flow so as to adjust the direction of the rotational axis in parallel to the direction of the stream. In this example, the structure rotates clockwise against the flow (from left to right). In contrast, a mirror-image structure rotates in the opposite direction. Simply put, it rotates in the slanted direction of the center area.

Another example regarding motion is provided in FIG. 10 . This is a rectangular twisted structure in FIG. 6 , wherein C=0 (B>0, Y=2A). When placed on a slope, this structure makes awkward movements by changing the direction of travel. 

1. A structure, having a three-dimensional shape that can be represented by a net in a two-dimensional shape, the two-dimensional shape of the net comprising: 2 congruent rotationally symmetric convex polygons, wherein the polygon is a n-gon and n is an integer more than 3; 2n three-valent main-vertices, wherein each of the main-vertices is placed at each corner of the polygons without overlapping; 2n three-valent sub-vertices, wherein n of the sub-vertices are placed at rotationally symmetric positions within each of the polygons without overlapping; 2n main-edges, wherein each of the main-edges is straight, and each of the main-edges connects 2 of the main-vertices without crossing; and 4n sub-edges, wherein each of the sub-edges is either straight or curved, each of the sub-edges connects two of the sub-vertices or connects one of the main-vertices and one of the sub-vertices, and 2n of the sub-edges are placed at rotationally symmetric positions within each of the polygons without crossing, wherein: each of the main-vertices contacts 2 of the main-edges and one of the sub-edges, and each of the sub-vertices contacts 1 main-edge and the 2 sub-edges, and wherein the structure has curved developable surfaces with a biaxially twisted appearance.
 2. The structure of claim 1, wherein the polygon is a regular polygon.
 3. The structure of claim 1, wherein the structure is a box and the net further comprises a plurality of flaps and/or crease lines.
 4. The structure of claim 1, wherein the structure is a container formed in the three-dimensional shape that can be represented by the net.
 5. The structure of claim 1, wherein the structure rotates in a flow about its rotational axis when a direction of the rotational axis is in parallel to a direction of the flow. 